Turns out that the actual RF signal being transmitted is straightforward analog FM.
A £10 USB TV stick and the free SDR# software http://www.rtl-sdr.com/rtl-sdr-quick-start-guide/ can decode the signal and play it out of the sound card output in stereo.

Shure mentions that there is some "24 bit digital" processing - what they call "Patented Shure Audio Reference Companding". The decoded PC signal does sound a bit swooshy compared to the actual Shure receiver unit, so that would suggest it is doing a bit of companding.